wave op• Data kinds: none → image2d (an op determined by its arguments alone — it takes no image or data input)
• Call: import optics; optics.airy_pattern(size=64, wavelength_um=0.55, f_number=5.6, pixel_pitch_um=0.5) (or opsoptics.get("airy_pattern"))
The diffraction-limited PSF of a circular pupil (Airy pattern).
`I(r) = [2*J1(v)/v]^2 with v = pi*r/(lambda*N), r` the radial
distance in the image plane, `N` the working f-number. Sampled on a
`size x size` grid centred between pixels for even *size* and on a pixel
for odd *size*. The normalisation is analytic (`I(0) = 1` by the
`v -> 0` limit), not a division by the sampled maximum: for odd *size*
the centre pixel is therefore exactly 1.0, and for even *size* the true
peak falls between pixels so the largest *sample* is below it (0.9679 at
`size = 8` with the defaults, measured). Rescaling to the sampled maximum
instead would quietly change the physics with the parity of the grid.
Returns a `(size, size)` float64 intensity image.
Ground truth it reproduces (measured, `tests/test_optics.py`): the first
dark ring sits at the first zero of `J1, r = 1.2197*lambda*N` — at
`lambda = 0.55 um, N = 5.6 that is 3.7567 um`, and the sampled
radial minimum lands at `3.760 um` on a 0.01 um grid (0.3 of a sample
away, which is the sampling, not an error); the peak is exactly 1.0 at the
centre and the pattern is symmetric to 1e-16.
The encircled energy inside that ring is the textbook 83.8% of the *whole
infinite* pattern — which a finite grid cannot measure: the Airy tails fall
off only as `1/r^3`, so a 25.6 um half-width grid reports 0.857 and a
51.2 um one 0.847 (both measured). The number is quoted here as physics,
not as something this op returns.
The `v -> 0` limit is evaluated explicitly as 1.0 rather than left to
`0/0`: that division is the classic silent-NaN in every hand-rolled Airy
routine, and the centre pixel is exactly where it bites.
Raises `ValueError: *size* outside [2, MAX_GRID]`; non-positive or
non-finite *wavelength_um*, *f_number*, *pixel_pitch_um*.
Scalar, aberration-free, unobstructed circular pupil, low NA. A central
obscuration (a mirror telescope) changes the ring structure; high NA needs a
vector treatment. For the *measured* PSF of a real system use
:func:psf_to_mtf on an image of a point source instead.
Every optics op validates its input before computing (nothing slips through silently):
• Units are baked into the argument name — _mm / _um / _deg / _mrad. Confusing mm with µm does not crash; it yields a plausible-looking wrong answer, so the name prevents it. Nothing here guesses the unit from the magnitude.
• **Strings raise ValueError** — float('50') succeeds, so an unparsed configuration value would slip through as a length (measured: thin_lens('50', '200') returned a plausible 66.667 mm). bool is refused too, as the implicit promotion True == 1.
• **complex / masked arrays raise ValueError (real-valued slots only; silently dropping the imaginary part or peeling off the mask is refused). NaN/Inf raises ValueError on every input.**
• Division by zero and its relatives are refused by name: focal length 0, radius of curvature 0, refractive index <= 0, a fully opaque aperture (all zeros, so the normalisation is 0/0), a PSF whose sum is <= 0, a Stokes vector with S0 = 0, and an object sitting at the front focal point (the image is at infinity).
• Only two ops return a non-finite value, and both state it as a contract: depth_of_field returns far_mm = inf beyond the hyperfocal distance (that is what the hyperfocal distance means), and gaussian_beam returns wavefront_radius_mm = inf at the waist (the radius of curvature of a plane wavefront). Both also return a finite companion (far_is_infinite / curvature_per_mm). **Any other silent NaN/Inf is detected internally and raises ValueError** — "float64 overflowed" and "the answer is infinite" are different claims, so the first is never returned wearing the face of the second.
• Size caps: generated grids are capped by optics.MAX_GRID (4096); supplied fields/PSFs/apertures by optics.MAX_FIELD_ELEMENTS (2^24); ABCD element chains by optics.MAX_SYSTEM_ELEMENTS (1024); Zernike by MAX_ZERNIKE_TERMS (512) / MAX_ZERNIKE_ORDER (40) / MAX_ZERNIKE_BASIS (2^25). This closes, fail-closed, the paths where a small argument triggers a huge internal allocation (measured: n_max=40 × 4096² needs 108 GB).
• Physically impossible states are refused too: a Stokes vector with degree of polarisation > 1, negative transmittance, negative intensity, and invalid Zernike indices such as n-|m| odd.
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• optics_imaging — py -3.11 examples/optics_imaging.py
image2d as input)fraunhofer_pattern · psf_to_mtf · illumination_uniformity · render_through_lens · surface_defect · defocus_blur
wave)angular_spectrum_propagate · fraunhofer_pattern · gaussian_beam
*Provenance: optics.py — OPTICS operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.